Problem: A traveler comes upon a fork in the road. On the path to the traveler's right, a sign reads "Mercer: $24\,\text{km}$." On the path to the traveler's left, a sign reads "Turtle Lake: $17\,\text{km}$." The traveler also observes that the angle between the paths is $75^\circ$. Assuming both paths are perfectly straight, what is the distance between Mercer and Turtle Lake? Do not round during your calculations. Round your final answer to the nearest tenth of a kilometer.
Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $AB=d$. $75^\circ$ $d$ $24\text{ km}$ $17\text{ km}\,\,\,$ $A$ $B$ $C$ Since we are given two side lengths and the angle measure between them, we can use the law of cosines. Using the law of cosines $\begin{aligned} (AB)^2&=(AC)^2+(BC)^2-2AC\!\cdot\! BC\!\cdot\!\cos(C)\\\\ d^2&=17^2+24^2-2\cdot 17\cdot 24\cdot\cos(75^\circ) \gray{\text{Substitute}}\\\\ d&=\sqrt{17^2+24^2-2\cdot 17\cdot 24\cdot\cos(75^\circ)}\\\\ d&\approx 25.6 \end{aligned}$ The answer The distance between Mercer and Turtle Lake is $25.6$ kilometers.